I had the chance to read Dray & Minogue's online stuff shortly before I was first assigned to teach a Calculus course, an Applied Calculus course intended for biology and economics majors. As it was a terminal math course, I felt justified in taking an unusual approach; as it was applied, I didn't worry about rigour (as long as I knew that it could be done). So I used differentials. It seemed to go all right.
That was three years ago. Since then, I've also taught the regular three-semester-equivalent Calculus series for physical science/engineering and mathematics majors. Sometimes I teach the whole sequence in a row, sometimes I only teach Caclulus 3 (multivariable). I have used differentials every time. I particularly love it for multivariable calculus, where I also introduce differential forms.
It is generally not too hard to link this to what's in the book (which is chosen at the departmental level). After all, differentials (and differential forms in multivariable integrals) appear in the text; it's just that they only appear in certain combinations. So I can pick out a subexpression from the expressions in the book and explain that it has a meaning in its own right. Sometimes I write some new exercises, but mostly I just show them another way to do the exercises in the book. Sometimes they do it my way, sometimes the book's way; it's all good.
In the applied course, I do differentials before derivatives, which I do before limits. We're not pretending to define things rigorously, so there is no harm in this, and this seems to be in order of increasing difficulty. I actually introduce rigorous definitions (even ones that are not in the textbook), but at the end, in the context of approximation methods (which is actually how they first appeared historically); I mention that such-and-such actually provides a rigorous definition, which they take as one of my remarks important for their well-rounded liberal education that won't be on the test, and we move on.
In the regular course, I feel obligated to give the rigorous definitions a more prominent place, since they are explicitly part of the syllabus. This means an approach as in Aeryk's answer: limits, then derivatives, then differentials. I pretty much agree with everything that Aeryk has said, right down the line. But I will add that differentials are especially awesome in multivariable calculus, where I can introduce partial derivatives as in David Butler's answer. Often it is easier to work with the differentials directly, and never mind the individual partial derivatives that appear within them.
I must caution against conflating different kinds of differentials and infinitesimals. In particular, Dray & Minogue's approach is NOT the same as Keisler's approach using nonstandard analysis. (Although a nice thing about Keisler's book is that it allows you to do derivatives before limits, rigorously even.) Dray & Minogue's (and hence my) differentials are the linear differentials from differential geometry (and in differential forms), not the nonstandard hyperreal numbers from nonstandard analysis (nor the nilpotent infinitesimals from synthetic differential geometry, for that matter). To distinguish these, consider: if ‘d’ indicates a finitesimal change, then dy/dx = f'(x) is an approximation; if it indicates a nonstandard infinitesimal, then dy/dx = f'(x) is an adequality (equality of standard parts); if it indicates a linear differential, then dy/dx = f'(x) is an equality, period. (All on the assumption that y = f(x), of course.)
I also want to stress the triviality of the Chain Rule with differentials. I don't mean that you can prove it by cancelling du in dy/dx = dy/du du/dx; it is a real theorem that requires a real proof. (I once read a review of Keisler's book that praised Keisler for being able to prove the Chain Rule by a trivial cancellation; I can only assume that the reviewer did not read that part of the book, for Keisler himself was under no such illusions.) But when working with differentials, it becomes a theoretical result, not a tool for practical calculation. The equation dy/dx = dy/du du/dx is a triviality, but it is not the Chain Rule; or rather, it is the Chain Rule only if you change the meaning of the symbols from one place to another, an unfair trick to play on students. The real Chain Rule is d(f(u)) = f'(u) du. It tells you how, if you can differentiate [take the derivative of] a function, you can apply that function to any differentiable expression and differentiate [take the differential of] the resulting expression.
An example will show the power of this approach. Suppose that you establish, say by the usual argument involving sum-angle formulas and special limits, that the derivative of the sin function is the cosine function. Then you apply the Chain Rule —once!— to conclude that d(sin u) = cos u du for any differentiable expression u; this fact is called the Sine Rule. Now when faced with such expressions as sin(3x²), sin(5x - sin x), sin(2x + 3y), etc, you do NOT use the Chain Rule; you use the Sine Rule. So in the end, instead of having a rule for each algebraic operation, a derivative for each special function, and a Chain Rule that requires you to analyse expressions as composites, you have a rule for each algebraic operation and a rule for each special function, and every differentiation is done by working your way from the outside in, applying the appropriate rule to whatever operation comes next in the expression as it is written down. (This is at least part of what Aeryk means by saying that Calculus becomes algebraic.) The Chain Rule is only needed explicitly if you are dealing with a new or unknown function.
As you can see with sin(2x + 3y) above, differentials work seamlessly with any number of variables. Students have done Algebra with multiple variables, and they can just as easily do Calculus with multiple variables. Some things, such as optimization, graphing, and integration, are legitimately more complicated with multiple variables, but the basic operation of differentiating [taking the differential of] an expression is not. This is why related rates and implicit differentiation comes so smoothly, but it also means that you can talk about partial derivatives right away too. I make sure to do this, even though it is again an aside that's not on the text, because Calculus 3 is not required for everybody who is going to use partial derivatives, after all.
You can see material for my courses at http://tobybartels.name/courses/. Generally the later material is more polished. All that you really need to look at are the notes at the bottom of each course's main page. You will need a DjVu reader to read some of the older stuff (which has been converted to this smaller file format only).